Oregon Mathematics Content Standards Specifications
Fluency Statement
What are the Main Messages of NCTM's Principles and Standards (2000) Regarding Computation?
|Computational fluency is an essential goal for school mathematics (p. 152): |
|Embedding Fluency in Conceptual Understanding |
|The methods that a student uses to compute should be grounded in understanding (pp. 152-55). |
|Students can achieve computational fluency using a variety of methods and should, in fact, be comfortable with more than one approach (p. 155). |
|Students should have opportunities to invent strategies for computing using their knowledge of place value, properties of numbers, and the operations (pp. 35 |
|and 220). |
|Students should investigate conventional algorithms for computing with whole numbers (pp. 35 and 155). |
|Goals of Fluency |
|Students should know the basic number combinations for addition and subtraction by the end of grade 2 and those for multiplication and division by the end of |
|grade 4 (pp. 32, 84, and 153). |
|Students should be able to compute fluently with whole numbers by end of grade 5 (pp. 35, 152, and 155). |
|Students should be encouraged to use computational methods and tools that are appropriate for the context and purpose, including mental computation, |
|estimations, calculators, and paper and pencil (pp. 36, 145, and 154). |
| |
What is Computational Fluency?
NCTM Principles and Standards of School Mathematics (2000) defines computational fluency as having efficient and accurate methods for computing that are based on well understood properties and number relationships.
The National Math Panel Report cites the NCTM definition of computational fluency in its report when it uses this phrase. For further clarity, on page 3-41 of the Task Group Reports of the National Mathematics Advisory Panel, there is a discussion of the critical foundations for the study of algebra: (1) fluency with whole numbers, (2) fluency with fractions, and (3) particular aspects of geometry and measurement.
The excerpt from page 3-41 is below:
1. Fluency with whole numbers
By the end of the elementary grades, children should have a robust sense of number. This sense of number must include understanding place value, and the ability to compose and decompose whole numbers. It must clearly include a grasp of the meaning of the basic operations of addition, subtraction, multiplication, and division, including use of the commutative, associative, and distributive properties; the ability to perform these operations efficiently; and the knowledge of how to apply the operations to problem solving. Computational facility rests on the automatic recall of addition and related subtraction facts, and of multiplication and related division facts. It requires fluency with the standard algorithms for addition, subtraction, multiplication, and division. Fluent use of the algorithms not only depends on the automatic recall of number facts but also reinforces it. A strong sense of number also includes the ability to estimate the results of computations and thereby to estimate orders of magnitude, e.g., how many people fit into a stadium, or how many gallons of water are needed to fill a pool.
2. Fluency with Fractions
Before they begin algebra course work, middle school students should have a thorough understanding of positive as well as negative fractions. They should be able to locate both positive and negative fractions on the number line; represent and compare fractions, decimals, and related percents; and estimate their size. They need to know that sums, differences, products, and quotients (with nonzero denominators) of fractions are fractions, and they need to be able to carry out these operations confidently and efficiently. They should understand why and how (finite) decimal numbers are fractions and know the meaning of percentages. They should encounter fractions in problems in the many contexts in which they arise naturally, for example, to describe rates, proportionality, and probability. Beyond computational facility with specific numbers, the subject of fractions, when properly taught, introduces students to the use of symbolic notation and the concept of generality, both being an integral part of Algebra (Wu, 2001).
3. Particular Aspects of Geometry and Measurement
Middle-grade experience with similar triangles is most directly relevant for the study of algebra: Sound treatments of the slope of a straight line and of linear functions depend logically on the properties of similar triangles. Furthermore, students should be able to analyze the properties of two- and three-dimensional shapes using formulas to determine perimeter, area, volume, and surface area. They should also be able to find unknown lengths, angles, and areas
|Core Standard: H.1A Algebra and Numeracy Score Reporting Category 1 |
|Demonstrate a deep understanding of real numbers and algebraic symbols by fluently creating, manipulating, computing with, and determining equivalent expressions, both numeric and symbolic. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.1A.1 Compare, order, and locate real numbers on a number |Items assessing this standard include locating real numbers on a number line,|Which number has the greatest value? |
|line. |comparing two real numbers, and ordering real numbers, i.e. smallest to | |
| |largest. |C. |
| |Problems may include a mixture of fractions, decimals (repeating and | |
|Assessable Academic Vocabulary: |non-repeating), pi, and/or square roots. |D. |
| | | |
|real numbers | | |
|number line | | |
|rational numbers | |Which number has the greatest value? |
|irrational numbers | | |
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| |Content Connections from Previous Grades: | |
| |4.1.4, 7.1 | |
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|Symbols and Notation: | | |
|[pic]”is not equal to” | | |
|< “is less than” | | |
|[pic]”is less than or equal to” | | |
|>”is greater than” | | |
|[pic]”is greater than or equal to” | | |
|[pic][pic]”pi” | | |
|Core Standard: H.1A Algebra and Numeracy Score Reporting Category 1 |
|Demonstrate a deep understanding of real numbers and algebraic symbols by fluently creating, manipulating, computing with, and determining equivalent expressions, both numeric and symbolic. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.1A.2 Evaluate, compute with, and determine equivalent |Items assessing this standard include performing computations with real |If a = [pic] and b = [pic], which of the following expressions gives |
|numeric and algebraic expressions with real numbers and |numbers, evaluating and simplifying expressions involving real numbers and/or|you the greatest value? |
|variables that may also include absolute value, integer |algebraic symbols, and determining if expressions are equivalent. | |
|exponents, square roots, pi, and/or scientific notation. |Real numbers and variables may include | |
| |absolute value | |
|Assessable Academic Vocabulary: |integer exponents | |
| |square roots | |
|evaluate |pi | |
|equivalent expressions |scientific notation. | |
|absolute value | | |
|integer exponents | | |
|square roots | | |
|pi | |(Evaluate an algebraic expression and may include absolute value.) |
|scientific notation | | |
| |Content Connections from Previous Grades: | |
| |7.1 |(Numeric expressions with integer exponents, or square roots, or |
|Symbols and Notation: | |scientific notation.) |
|[pic][pic]”pi” | |ITS: 100593 |
|[pic] “absolute value of 3” | | |
|Core Standard: H.1A Algebra and Numeracy Score Reporting Category 1 |
|Demonstrate a deep understanding of real numbers and algebraic symbols by fluently creating, manipulating, computing with, and determining equivalent expressions, both numeric and symbolic. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.1A.3 Express square roots in equivalent radical form and |Items assessing this standard include writing square roots in simplest |(Write in simplest radical form, or determine if two radical |
|their decimal approximations when appropriate. |radical form. The square roots may include variables as well as real |expressions are equivalent, or give a decimal approximation for a |
| |numbers. |square root.) |
| |When appropriate, calculators may be used to find decimal approximations for | |
|Assessable Academic Vocabulary: |square roots. | |
| | | |
|square root | | |
|radical sign | | |
|simplify | |(Simplify, or the area of a square is 12, what is the |
|simplest radical form | |length of the side?) |
| | | |
| | | |
| | | |
|Symbols and Notation: |Content Connections from Previous Grades: | |
|[pic]”radical sign” |8.3 |(Write in simplest radical form, or better yet in some |
|[pic] ”square root of 25” | |context, i.e. using Pythagorean theorem.) |
| | | |
|Core Standard: H.1A Algebra and Numeracy Score Reporting Category 1 |
|Demonstrate a deep understanding of real numbers and algebraic symbols by fluently creating, manipulating, computing with, and determining equivalent expressions, both numeric and symbolic. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.1A. 4 Develop, identify, and/or justify equivalent algebraic|Items assessing this standard must include using one or more of the following|(Give an expression and have students identify the simplified |
|expressions, equations, and inequalities using the properties |properties to justify two algebraic expressions or equations are equivalent: |expression.) |
|of exponents, equality and inequality, as well as the |properties of exponents | |
|commutative, associative, inverse, identity, and distributive |properties of equality | |
|properties. |properties of inequality | |
| |commutative property |(Use properties of inequalities.) |
|Assessable Academic Vocabulary: |associative property | |
|equivalent expressions |inverse property | |
|equivalent equations |identity property | |
|inequalities |distributive property | |
|properties of exponents |Expressions may include variables and/or real numbers. |(Justify that two algebraic expressions or equations are equivalent |
|properties of equality |Students may be asked to simplify an algebraic expression using any of the |using a combination of properties of exponents.) |
|properties of inequality |properties listed above. | |
|commutative property | | |
|associative property | | |
|inverse property | | |
|identity property |Content Connections from Previous Grades: | |
|distributive property |6.3, 7.1.4 | |
| | | |
|Symbols and Notation: | | |
|(n2)3 “n squared, raised to the third power” | | |
|>”is greater than” | | |
|[pic]”is greater than or equal to” | | |
|[pic]”is not equal to” | | |
|< “is less than” | | |
|[pic]”is less than or equal to” | | |
|Core Standard: H.1A Algebra and Numeracy Score Reporting Category 1 |
|Demonstrate a deep understanding of real numbers and algebraic symbols by fluently creating, manipulating, computing with, and determining equivalent expressions, both numeric and symbolic. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H1A.5 Factor quadratic expressions limited to factoring |Items assessing this standard include factoring quadratics of the form |(Factor an expression by using a common monomial term.) |
|common monomial terms, perfect-square trinomials, differences |ax2 + bx + c, where a=1, and the quadratic factors over the integers. | |
|of squares, and quadratics of the form x2 + bx + c that factor |Students will also be asked to factor perfect-square trinomials, difference | |
|over the integers |of squares, and common monomial terms in quadratic expressions that may not | |
| |have a leading coefficient of 1. | |
|Assessable Academic Vocabulary: |Problems may include justifying solutions by multiplying. |(Factor perfect-square trinomials. (Or factor a quadratic where “b” |
| | |and “c” are both positive)) |
|factor | | |
|greatest common factor | | |
|difference of squares | | |
| | | |
| | | |
| |Content Connections from Previous Grades: |(Factor difference of squares. (Or factor a quadratic where “b” and |
| |6.3 |“c” have different signs)) |
|Symbols and Notation: | | |
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|Core Standard: H.2A Algebra Score Reporting Category 1 |
|Use linear equations and functions to represent relationships and solve linear equations, linear inequalities, systems of linear equations, and systems of linear inequalities. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries for Assessable Content: |Sample Items: |
|H.2A.1 Identify, construct, extend, and analyze linear |Items assessing this standard include the following: recognize, describe, |(Extend a linear pattern or table of values for a linear function to |
|patterns and functional relationships that are expressed |extend, and analyze linear functions and patterns in a variety of ways: |find the nth term.) |
|contextually, numerically, algebraically, graphically, in |numerically (e.g., 4, 7, 10, …) | |
|tables, or using geometric figures. |algebraically (e.g., 3x + 4 or y = 3x + 4 or f(x) = 3x + 4) | |
| |graphically (e.g., a line drawn on a coordinate plane) | |
|Assessable Academic Vocabulary: |geometrically (e.g., show shapes that increase in a linear pattern like one | |
| |square, three squares, five squares, etc) | |
|numerically |contextually (e.g., a word problem) |(Given a graph, describe a possible context or situation or vice |
|algebraically | |versa.) |
|extend | | |
|linear pattern | | |
| | | |
| | | |
| | | |
|Symbols and Notation: | |(Match a table of values with the representation of the same function |
| |Content Corrections from Previous Grades: |using geometric shapes. (Or analyze a pattern from a table that does |
| |6.3, 8.1.1 |not have sequential values and identify the equation for the table of |
| | |values.)) |
|Core Standard: H.2A Algebra Score Reporting Category 1 |
|Use linear equations and functions to represent relationships and solve linear equations, linear inequalities, systems of linear equations, and systems of linear inequalities. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.2A.2 Given a rule, a context, two points, a table of |Items assessing this standard include identifying the slope of a line, |(Given a rule, a context, two points, or linear equations in slope |
|values, a graph, or a linear equation in either slope intercept|determining the x and/or y intercept(s), and interpreting the meaning of each|intercept form, identify the slope.) |
|or standard form, identify the slope of the line, determine the|given one of the following: | |
|x and/or y intercept(s), and interpret the meaning of each. |a rule | |
| |a context | |
| |two points | |
|Assessable Academic Vocabulary: |a table of values | |
| |a graph | |
|slope |a linear equation in slope intercept form |(Given a graph, identify the slope, intercepts, and meaning of each.) |
|x-intercept |a linear equation in standard form | |
|y-intercept | | |
|slope-intercept form: y=mx+b | | |
|standard form: Ax+By=C | | |
| | | |
| | |(Given a linear equation in standard form, identify the slope of the |
|Symbols and Notation: | |line, x and/or y intercept(s) and interpret the meaning of each.) |
| |Content Connections from Previous Grades: | |
| |6.3.5, 7.2.1, 8.1 | |
| | | |
|Core Standard: H.2A Algebra Score Reporting Category 1 |
|Use linear equations and functions to represent relationships and solve linear equations, linear inequalities, systems of linear equations, and systems of linear inequalities. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.2A.3 Determine the equation of a line given any of the |Items assessing this standard include determining the equation of a line |Line A is the graph for the equation [pic]. Line B is perpendicular to|
|following information: two points on the line, its slope and |given any of the following: |line A. |
|one point on the line, or its graph. Also, determine an |two points on the line |What is the equation of the graph of line B? |
|equation of a new line, parallel or perpendicular to a given |a slope and one point on the line |[pic] |
|line, through a given point. |a graph | |
| |a parallel line and a given point | |
|Assessable Academic Vocabulary: |a perpendicular line and a given point | |
| |Linear equations may be written in slope-intercept form or standard form. | |
|equation of a line |Students may be asked to find the equation for a line parallel or | |
|parallel |perpendicular to a given line through a point, but the line referenced may | |
|perpendicular |only be represented by two points rather than an equation. |ITS #100398 |
| | |(Given two points on a line or the graph of a line, write the equation |
| | |of a line parallel to this given line through a point.) |
| | | |
| |Content Connections from Previous Grades: |(Given the graph of a line, write the equation of the line |
| |8.1 |perpendicular to this line at a given point.) |
|Symbols and Notation | | |
| | | |
|Core Standard: H.2A Algebra Score Reporting Category 1 |
|Use linear equations and functions to represent relationships and solve linear equations, linear inequalities, systems of linear equations, and systems of linear inequalities. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.2A.4 Fluently convert among representations of linear |Items assessing this standard include converting fluently among one or more |(Given an equation of a line in slope-intercept form, graph its line.) |
|relationships given in the form of a graph of a line, a table |of the following representations of linear relationships: | |
|of values, or an equation of a line in slope-intercept and |graph of a line | |
|standard form. |table of values | |
| |equation written in slope-intercept form | |
|Assessable Academic Vocabulary: |equation written in standard form | |
| | |(Given a table of values, write the equation of the line in |
|slope-intercept form: y=mx+b | |slope-intercept form.) |
|standard form: Ax+By=C | | |
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| | | |
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|Symbols and Notation | |(Given a graph, write the equation of the line in standard form.) |
| |Content Connections from Previous Grades: | |
| |8.1.1, 8.1.2, 8.1.3 | |
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|Core Standard: H.2A Algebra Score Reporting Category 1 |
|Use linear equations and functions to represent relationships and solve linear equations, linear inequalities, systems of linear equations, and systems of linear inequalities. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.2A.5 Given a linear function, interpret and analyze the |Identify the relationship between the independent and dependent variables of |(Given a function, identify and describe the relationship between the |
|relationship between the independent and dependent variables. |a linear function. |independent variable and the dependent variable.) |
|Solve for x given f(x) or solve for f(x) given x. |Analyze the relationship between the independent and dependent variables, | |
| |given a linear function in a table, context or graph. | |
|Assessable Academic Vocabulary: |Solve for f(x) given x. | |
| |Solve for x given f(x). |(Solve for f(x) given x.) |
|independent variable | | |
|dependent variable | | |
| | | |
| | | |
| | |(Solve for x given f(x).) |
| | | |
|Symbols and Notation |Content Connections from Previous Grades: | |
| |6.3.3, 7.1.4, 8.1, H.1A, H.2A.2, H.2A.4 | |
|[pic] “f of x” | | |
| | | |
|Core Standard: H.2A Algebra Score Reporting Category 1 |
|Use linear equations and functions to represent relationships and solve linear equations, linear inequalities, systems of linear equations, and systems of linear inequalities. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.2A.6 Analyze how changing the parameters transforms the |Items assessing this standard include analyzing transformations of the graph |(Given a linear equation in the form |
|graph of |of |[pic] |
|[pic] |[pic] |What happens when b changes? |
| | | |
|Assessable Academic Vocabulary: |Explain how changes for the values of “m” and “b” affect the graph of [pic] |What happens when m changes? |
| |How is the graph affected by changes in m? | |
|transform |How is the graph affected by changes in b? |What happens when m and b are changed simultaneously? |
| |The equation for the graph may be written in function form or slope-intercept| |
| |form. |What happens when m and b are changed simultaneously by the same |
| | |amount? |
| | | |
| | |Given a “transformed” graph, determine the old and/or new function.) |
| |Content Connections from Previous Grades: | |
| |8.1.2 | |
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|Symbols and Notation | | |
| | | |
|[pic] “f of x” | | |
| | | |
|Core Standard: H.2A Algebra Score Reporting Category 1 |
|Use linear equations and functions to represent relationships and solve linear equations, linear inequalities, systems of linear equations, and systems of linear inequalities. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.2A.7 Write, use, and solve linear equations and |Items assessing this standard include |You can receive a reduced ticket price at the local movie theater if |
|inequalities using graphical and symbolic methods with one or |writing, using, and solving linear equations and inequalities using graphical|you are 65 or older, or if you are younger than 12. |
|two variables. Represent solutions on a coordinate graph or |and symbolic methods with |Which graph displays this information? |
|number line. |one variable. |[pic] |
| |two variables. |[pic] |
| |Represent solutions with one variable on a number line. |[pic] |
|Assessable Academic Vocabulary: |Represent solutions with two variables on a coordinate graph. |[pic] |
| | |ITS #100566 |
| | |The graph shows the solution to which of these inequalities? |
| | |[pic] |
| | |y > 3x + 3 |
| | |y < 3x + 3 |
| | |y > x + 3 |
| |Content Connections from Previous Grades: 7.1.4, 8.1.4 |y < x + 3ITS #100426 |
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|Symbols and Notation | | |
|[pic]”is not equal to” | | |
|< “is less than” | | |
|[pic]”is less than or equal to” | | |
|>”is greater than” | | |
|[pic]”is greater than or equal to” | | |
|Core Standard: H.2A Algebra Score Reporting Category 1 |
|Use linear equations and functions to represent relationships and solve linear equations, linear inequalities, systems of linear equations, and systems of linear inequalities. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.2A.8 Solve systems of two linear equations graphically and |Solve systems of two linear equations graphically. |Find the solution to the following system of two linear equations: |
|algebraically, and solve systems of two linear inequalities |Solve systems of two linear equations algebraically which may include using |[pic] |
|graphically. |substitution, elimination, or matrices. |[pic] |
| |Solve systems of two linear inequalities |[pic] |
| |graphically. | |
|Assessable Academic Vocabulary: | |[pic] |
| | | |
|system of linear equations | |[pic] |
|system of linear inequalities | | |
| | |[pic] |
| | | |
| | |ITS #100602 |
| |Content Connections from Previous Grades: |(Solve systems of two linear equations algebraically.) |
|Symbols and Notation |8.1.5, 8.1.6 | |
|< “is less than” | |(Solve systems of two linear inequalities graphically.) |
|[pic]”is less than or equal to” | | |
|>”is greater than” | | |
|[pic]”is greater than or equal to” | | |
|Core Standard: H.3A Algebra Score Reporting Category 1 |
|Use quadratic and exponential equations and functions to represent relationships. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.3A.1 Given a quadratic or exponential function, identify or |Items assessing this standard give students a quadratic or exponential |(Given a quadratic function, identify a corresponding table.) |
|determine a corresponding table or graph. |function. Students have to identify a corresponding table or graph or | |
| |determine a corresponding table or graph. | |
| |Quadratic and exponential functions may be expressed using function notation | |
|Assessable Academic Vocabulary: |or y=. | |
| | | |
|quadratic function | | |
|exponential function | |(Determine the graph of an exponential function showing decay.) |
| | | |
| | | |
| | | |
| | | |
| | |(Given a mixture of quadratic and exponential functions, determine the |
|Symbols and Notation: | |corresponding table or graph.) |
| |Content Connections from Previous Grades: | |
| |8.1 | |
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| | | |
|Core Standard: H.3A Algebra Score Reporting Category 1 |
|Use quadratic and exponential equations and functions to represent relationships. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.3A.2 Given a table or graph that represents a quadratic or |Items assessing this standard use tables or graphs that represent a quadratic|(Given a table representing a quadratic function, extend the pattern to|
|exponential function, extend the pattern to make predictions. |or exponential function and extend the pattern to make predictions. |find the nth term.) |
| |The independent variable values in the table will have a constant difference | |
| |(e.g. -2, -1, 0, 1, 2, … or 2, 4, 6, … etc) | |
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|Assessable Academic Vocabulary: | | |
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| | | |
| | |(Given a graph representing an exponential function, extend the pattern|
| | |to make predictions.) |
| | | |
| |Content Connections from Previous Grades: | |
| |8.1.4 | |
| | |(Given tables and/or graphs that represent quadratic and/or exponential|
|Symbols and Notation: | |function, extend the pattern to see which one reaching a certain amount|
| | |sooner over a given period of time.) |
|Core Standard: H.3A Algebra Score Reporting Category 1 |
|Use quadratic and exponential equations and functions to represent relationships. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.3A.3 Compare the characteristics of and distinguish among |Items assessing this standard use any of the following to compare and |This graph is an example of what type of function? |
|linear, quadratic, and exponential functions that are expressed|distinguish among linear, quadratic, and exponential functions: |[pic] |
|in a table of values, a sequence, a context, algebraically, |a table of values |Absolute Value |
|and/or graphically, and interpret the domain and range of each |a sequence |Exponential |
|as it applies to a given context. |a context |Linear |
| |an equation |Quadratic |
|Assessable Academic Vocabulary: |a graph | |
| | | |
|exponential |Identify and interpret the domain and range of linear, quadratic and/or |ITS #100062 |
|linear |exponential functions as it applies to a given context. | |
|quadratic |Equations may be written using f(x) or y =. |(Given a table of values, determine if the function is linear, |
|domain | |quadratic or exponential.) |
|range | | |
| | | |
| |Content Connections from Previous Grades: |(Given a context, determine if the function is linear, quadratic or |
| | |exponential. Interpret the domain and range.) |
|Symbols and Notation: | | |
|0< b < 1 | | |
|[pic] “0 is less than b is less than 1” | | |
| | | |
|Core Standard: H.3A Algebra Score Reporting Category 1 |
|Use quadratic and exponential equations and functions to represent relationships. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.3A.4 Given a quadratic or exponential function, interpret |Items assessing this standard use quadratic or exponential functions to do |A ball is tossed into the air. The height of the ball as a function of |
|and analyze the relationship between the independent and |the following: |time can be described by the equation [pic]. In this equation h is the |
|dependent variables, and evaluate the function for specific |Interpret and analyze the relationship between the independent and dependent |height of the ball in feet and t is time in seconds. |
|values of the domain. |variables. This may include determining for which values of x the y values |After how many seconds will the ball hit the ground? |
| |are increasing or decreasing. It also includes identifying the vertex, |4 seconds |
|Assessable Academic Vocabulary: |intercepts, or other important point of the function. |4.5 seconds |
| |Evaluate the function for specific values of the domain. A student may also |9 seconds |
|independent variable |need to solve the function to find a specific value in the domain. |56 seconds |
|dependent variable |Functions may be written using f(x)= or y=. . | |
| | |ITS #100138 |
| | | |
| | |(Given a quadratic function, describe where the turning point is.) |
| | | |
| | | |
|Symbols and Notation: | |(Given an exponential function, determine for what values of the domain|
| |Content Connections from Previous Grades: |the function increasing (or decreasing).) |
| | | |
| | | |
| | | |
|Core Standard: H.3A Algebra Score Reporting Category 1 |
|Use quadratic and exponential equations and functions to represent relationships. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.3A.5 Given a quadratic equation of the form [pic] with |Items assessing this standard use a quadratic equation of the form [pic] with|(Given a quadratic equation, find the roots by factoring.) |
|integral roots, determine and interpret the roots, the vertex |integral roots to do the following: | |
|of the parabola that is the graph of [pic], and an equation of |Determine and interpret the roots. This may be done by factoring, using the | |
|its axis of symmetry graphically and algebraically. |quadratic formula, or graphing. | |
| |Determine and interpret the vertex of the parabola (graph of [pic]). This |(Given a quadratic equation, determine and interpret the roots |
|Assessable Academic Vocabulary: |may be done using [pic]to find the x-coordinate and then using substitution |graphically.) |
| |to find the y-coordinate or by graphing. | |
|roots |Determine and interpret the axis of symmetry using [pic] or the graph of the | |
|x-intercepts |parabola. | |
|vertex | | |
|axis of symmetry | |(Given a quadratic equation, determine and interpret the vertex of the |
|parabola |Content Connections from Previous Grades: |parabola and an equation of its axis of symmetry.) |
| | | |
|Symbols and Notation: | | |
| | | |
|Core Standard: H.1G Geometry Score Reporting Category 2 |
|Apply properties of two-dimensional figures. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.1G.1 Identify, apply, and analyze angle relationships among|Items assessing this standard use alternate interior angles, alternate |(Identify two lines as parallel when consecutive interior angles are |
|two or more lines and a transversal to determine if lines are |exterior angles, corresponding angles, and same-side interior angles (also |60° and 120°.) |
|parallel, perpendicular, or neither. |named consecutive interior angles) between two lines and a transversal to | |
| |determine if lines are parallel, perpendicular, or neither. | |
|Assessable Academic Vocabulary: |Write and solve algebraic equations from angles given as expressions to | |
|parallel |determine if pairs of angles are congruent or supplementary. | |
|perpendicular | |(Identify two lines as perpendicular given a linear pair with |
|transversal | |expressions involving x.) |
|alternate interior angles | | |
|alternate exterior angles | | |
|corresponding angles | | |
|same-side interior angles | | |
|consecutive interior angles |Content Connections from Previous Grades: 8.3.1 | |
|supplementary | |(Give expressions with x for two alternate exterior angles and give a |
| | |linear pair angle to one of the exterior angles of 70°. Students will |
|Symbols and Notation: | |set up and solve for x using the linear pair and then substitute the |
|‖ “parallel” | |value for x into the other alternate interior angle to determine if |
|[pic] “perpendicular” | |lines are parallel.) |
|Arrows for parallel lines | | |
| | | |
|Right angle | | |
|Congruent angle symbols | | |
| | | |
| | | |
|or | | |
|Core Standard: H.1G Geometry Score Reporting Category 2 |
|Apply properties of two-dimensional figures. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.1G.2 Apply theorems, properties, and definitions to |Items assessing this standard use theorems, properties, and definitions to |Quadrilateral ABCD, with diagonal [pic], has |
|determine, identify, and justify congruency or similarity of |determine, identify, and justify congruent triangles. |[pic] . |
|triangles and to classify quadrilaterals. |determine, identify, and justify similar triangles |Which statement would make ΔBAD ΔDCB? |
| |classify quadrilaterals |[pic] C [pic] A |
|Assessable Academic Vocabulary: |Write and solve algebraic equations from expressions in triangles to | |
|congruent triangles |determine if triangles are similar or congruent. | |
|SSS |Write and solve algebraic equations from expression in figures to classify | |
|SAS |quadrilaterals. | |
|ASA | | |
|AAS | | |
|HL | | |
|similar triangles | |ITS #100594 |
|scale factor | | |
|classify |Content Connection from Previous Grades: |(Give two triangles with SAS marked and have them choose SAS for why |
|quadrilateral |3.3, 5.3.1, 6.2.1, 7.2.4 |triangles are congruent) |
|Symbols and Notation: | | |
|Congruency marks for sides and angles: | | |
| | |(Give a parallelogram – not to scale – with four expressions in terms |
| | |of x – a student has to solve for x by summing to 360 degrees and then |
| | |see that opposite angles are congruent) |
|Or | | |
| | | |
|Parallel sides marked with arrows: | | |
| | | |
| | | |
|Core Standard: H.1G Geometry Score Reporting Category 2 |
|Apply properties of two-dimensional figures. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.1G.3. Apply theorems of corresponding parts of congruent |Items assessing this standard solve for missing sides and/or angles using |[pic]CAT is similar to [pic]DOG. |
|and similar figures to determine missing sides and angles of |corresponding parts in |What is the measure of [pic]? |
|polygons. |congruent polygons. |[pic] |
| |similar polygons. |16 |
|Assessable Academic Vocabulary: |Write and solve algebraic equations from expressions for corresponding parts |18 |
| |in congruent or similar polygons to solve for a variable or a side or angle |20 |
|Corresponding Parts of Congruent Triangles are Congruent |measurement. |24 |
|(CPCTC) | | |
| | |ITS #100586 |
|congruent | | |
|congruent figures | |(Give two congruent pentagons, one with angle measures given, and |
|similar | |student finds corresponding measure in second pentagon.) |
|similar figures | | |
| |Content Connections from Previous Grades: 3.3, 6.2.1, 7.2.4 |(Give student two similar hexagons – give one angle as (x2 – x)° and |
|Symbols and Notation: | |the corresponding angle as (9x)°.) |
| | | |
|Congruency marks for sides and angles: | | |
| | | |
| | | |
| | | |
|Or | | |
| | | |
|Parallel sides marked with arrows: | | |
| | | |
| | | |
|Core Standard: H.1G Geometry Score Reporting Category 2 |
|Apply properties of two-dimensional figures. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.1G.4 Use trigonometric ratios (sine, cosine and tangent) |Items assessing this standard solve for missing sides in right triangles |Ted wants to measure the height of the football stands. He walks to |
|and the Pythagorean Theorem to solve for unknown lengths in |using sine, cosine, tangent, and/or the Pythagorean Theorem. |point A 250 feet from the stands. He measures the angle from the ground|
|right triangles. |Write the sine of an angle as the ratio of its opposite leg to its |to the top of the stands to be 17.75°. |
| |hypotenuse. |How high are the football stands? |
|Assessable Academic Vocabulary: |Write the cosine of an angle as the ratio of its adjacent leg to its |[pic] |
| |hypotenuse. |76 feet C. 238 feet |
|sine |Write the tangent of an angle as the ratio of its opposite leg to its |80 feet D. 781 feet |
|cosine |adjacent leg. | |
|tangent |Answers may be exact or approximate. |ITS #100367 |
|Pythagorean Theorem | |In circle A the radius of the circle is 17, CD = 16 and [pic] is |
| | |perpendicular to [pic]. |
| | |Use the information to find the length of [pic]. |
| | |[pic] |
|Symbols and Notation: | |5.7 C. 15.0 |
| | |8.0 D. 16.0 |
|sin A “sine of angle A” |Content Connections from Pervious Grades: 8.3.4, 8.3.5 |ITS #100448 |
|cos 40° “cosine of 40 degrees” | | |
|tan B “tangent of angle B” | | |
| | | |
| | | |
|Core Standard: H.1G Geometry Score Reporting Category 2 |
|Apply properties of two-dimensional figures. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.1G.5 Determine the missing dimensions, angles, or area of |Items assessing this standard use formulas, trigonometry, the Pythagorean |The area of the shaded region is ______. |
|regular polygons, quadrilaterals, triangles, circles, composite|Theorem, ratios, and properties of polygons and circles to determine missing |[pic] |
|shapes, and shaded regions. |dimensions, angles or area. | |
| |Use area formulas and/or proportions to solve for area of shaded regions. | |
|Assessable Academic Vocabulary: |Write and solve algebraic equations to find missing dimensions. | |
| |Apply properties of polygons and circles. | |
|regular polygon | |ITS #100603 |
|composite shape | | |
|region | |A roller coaster has a large rise and drop followed by a complete |
| | |circle. The following diagram shows measurements for the track. An |
| | |extra 20 feet are needed for cutting and welding. |
|Symbols and Notation: |Content Connections from Previous Grades: 3.3, 4.3, 5.3.2, 7.3, 8.3 | |
| | |How many feet of track should be ordered? (Use π = 3.14) |
| | |[pic] |
| | |280 feet |
| | |407 feet |
| | |415.6 feet |
| | |1537.4 feet |
| | |ITS #100606 |
|Core Standard: H.1G Geometry Score Reporting Category 2 |
|Apply properties of two-dimensional figures. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.1G.6 Determine if three given lengths form a triangle. If |Items assessing this standard ask students to use the triangle inequality |(Given three lengths – is the figure a triangle?) |
|the given lengths form a triangle, classify it as acute, right,|theorem and an extension of the converse of the Pythagorean Theorem. | |
|or obtuse. |Determine if three sides form a triangle. | |
| |Determine the range of lengths for the third side of a triangle given two | |
|Assessable Academic Vocabulary: |sides of the triangle. | |
| |Determine if a triangle is acute, right, or obtuse given three lengths. |(Given three lengths of a triangle, determine if the triangle is acute,|
|acute triangle | |right, or obtuse) |
|right triangle | | |
|obtuse triangle | | |
| | | |
|Symbols and Notation: | | |
| |Content Connections from Previous Grades: 3.3.1, 5.3.1, 8.3.4 | |
| | | |
| | |(Given two sides of a triangle, find the range of lengths for the third|
| | |side.) |
|Core Standard: H.1G Geometry Score Reporting Category 2 |
|Apply properties of two-dimensional figures. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.1G.7 In problems involving circles, apply theorems and |Items assessing this standard ask students to find lengths of chords, find |(Student knows tangent line to circle at radius makes a 90 degree |
|properties of chords, tangents, and angles; and theorems and |values for angles or lengths using tangents, and find angle measures inside |angle) |
|formulas of arcs and sectors. |and outside a circle. Students will use formulas and/or proportions to find | |
| |lengths of arcs and areas of sectors. | |
|Assessable Academic Vocabulary: |Find area of a circle. | |
| |Find circumference of a circle. | |
|chord |Write and solve algebraic equations using expressions for lengths of chords | |
|tangent |or measures of angles. | |
|inscribed angle |Answers may be exact or approximate. |(Student finds the degree of an inscribed angle given the degree |
|circumscribed angle | |measure of the intercepted arc) |
|arc | | |
|sector | | |
| | | |
|Symbols and Notation: |Content Connections from Previous Grades: 6.2.5, 7.3 | |
| | |(Student finds the area of a sector of a circle with a degree measure |
|AB “arc AB” | |of 50 degrees – or a degree measure that does not divide evenly into |
| | |360 degrees) |
|Core Standard: H.2G Geometry Score Reporting Category 2 |
|Apply properties of three-dimensional solids. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.2G.1 Identify, classify, model, sketch, and label |Items assessing this standard require students to know two-dimensional shapes|(Identify a net for a square pyramid) |
|representations of three-dimensional objects from nets and from|used in nets as well as the ability to identify a three-dimensional figure | |
|different perspectives. |given its net or views from different perspectives. | |
| |Identify views of a three-dimensional shape from different perspectives. | |
| |Label a three-dimensional shape given labels on its net. | |
| |Classify two- and three-dimensional figures. |(Show a cylinder with circles on top and bottom and ask for what the |
|Assessable Academic Vocabulary: | |side view looks like) |
| | | |
|net | | |
|three-dimensional | | |
| | | |
| | |(Given different perspectives of a triangular prism – top as a |
| | |triangle, two sides as rectangles, identify it as a triangular prism) |
|Symbols and Notation: | | |
| |Content Connections from Previous Grades: | |
| |3.3, 5.3, 7.3 | |
|Core Standard: H.2G Geometry Score Reporting Category 2 |
|Apply properties of three-dimensional solids. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.2G.2 Identify and apply formulas for surface area and |Items assessing this standard require students to find surface area and |When an elastic toy ball is fully inflated, it has a diameter of 7 |
|volume of spheres; right solids, including rectangular prisms |volume of three-dimensional figures. Students may need to find a missing |inches. If some of the air is removed, the diameter is only 5 inches. |
|and pyramids; cones; and cylinders; and compositions thereof. |dimension to calculate the surface area or volume. |What is the DIFFERENCE in the volume between the two sizes? |
|Solve related context-based problems. |Determine whether a word problem requires surface area or volume for its | |
| |solution. | |
|Assessable Academic Vocabulary: |Be able to understand formulas for surface area and volume. | |
| |Answers may be exact or approximate. | |
|surface area | |ITS #100441 |
|volume | | |
|sphere | |What is the volume left in the cylinder after the shaded cone region is|
|prism | |removed? |
|pyramid | |[pic] |
|cone | |1,024π in³ |
|cylinder |Content Connections from Previous Grades: 3.3, 5.3, 7.3 |1,536π in³ |
|lateral area | |2,048π in³ |
|base area | |4,096π in³ |
|slant height | |ITS #100266 |
|height | | |
| | | |
|Symbols and Notation: | | |
| | | |
|Formula sheet | | |
|Core Standard: H.2G Geometry Score Reporting Category 2 |
|Apply properties of three-dimensional solids. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.2G.3 Identify and apply formulas to solve for the missing |Items assessing this standard require students to solve for missing |(Find the slant height of a cone given its height and radius) |
|dimensions of spheres and right solids, including rectangular |dimensions of three-dimensional figures. Students may need to solve for the | |
|prisms and pyramids, cones, and cylinders, both numerically and|missing dimensions using trigonometry, the Pythagorean Theorem, or a formula | |
|symbolically. |involving a given surface area or volume. | |
| |Be able to understand formulas for surface area and volume. | |
|Assessable Academic Vocabulary: |Write and solve algebraic equations. | |
| |Answers may be exact or approximate. |(Find the radius of a sphere given its surface area … or … find the |
|surface area | |height of a cylinder given its volume) |
|volume | | |
|sphere | | |
|prism | | |
|pyramid | | |
|cone | | |
|cylinder |Content Connections from Previous Grades: 3.3, 5.3, 7.3 | |
|lateral area | |(Find the radius of a cylinder given its height and surface area…or… |
|base area | |find the radius of a sphere given its volume) |
|slant height | | |
|height | | |
| | | |
|Symbols and Notation: | | |
| | | |
|Formula sheet | | |
|Core Standard: H.2G Geometry Score Reporting Category 2 |
|Transform and analyze figures. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.3G.1 Recognize and identify line and rotational symmetry of|Items assessing this standard require students to identify two-dimensional |(Which quadrilateral has exactly one line of symmetry? Give square, |
|two-dimensional figures. |figures with rotational symmetry or line(s) of symmetry. Students may have |parallelogram, isosceles trapezoid, rectangle) |
| |to draw the line(s) of symmetry, determine the number of lines of symmetry, | |
| |or determine the measure of the rotational symmetry in a figure. | |
| |Identify and recognize line and rotational symmetry for polygons given in | |
|Assessable Academic Vocabulary: |words (e.g. rhombus). | |
| | |(Give a figure with 90 degree rotational symmetry and ask for the |
|line symmetry | |degree measure of the rotational symmetry) |
|rotational symmetry | | |
| | | |
| | | |
| | | |
|Symbols and Notation: | | |
| | |(Give four pictures…which one has 60 degree rotational symmetry?) |
| |Content Connections from Previous Grades: 3.3.8 | |
| | | |
| | | |
|Core Standard: H.3G Geometry Score Reporting Category 2 |
|Transform and analyze figures. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.3G.2 Identify and perform single and composite |Items assessing this standard require students to identify a given |The coordinates of point A are (-5, 3). |
|transformations of geometric figures in a plane, including |transformation using by identifying the picture of the image or the formula |If A is reflected over the y-axis, then translated 3 units right and 4 |
|translations, origin-centered dilations, reflections across |used to create the image from the preimage. Students may have to perform a |units down, the new coordinates of A are: |
|either axis or y = ±x, and rotations about the origin in |single or composite transformation to answer questions about the ordered |(5, 3) |
|multiples of 90˚. |pairs or picture of the image. |(8, -1) |
| |Be able to graph ordered pairs and connect them to make a geometric figure. |(-2, -1) |
|Assessable Academic Vocabulary: |Know the equations for the lines that are the x-axis and y-axis. |(-2, -7) |
| |Know translations, reflections, and rotations have congruent preimages and |ITS #100348 |
|transformation |images. | |
|translation |Know that dilations have similar preimages and images. Able to apply scale | |
|reflection |factor to describe and identify dilations. | |
|rotation |Understand how to apply composite transformations to a preimage. |(Give coordinates for a triangle ABC. Ask which ordered pair is for B’ |
|dilation | |after a dilation with a scale factor of 2/3… or … ask which ordered |
|scale factor | |pair is for B’ after a translation) |
|preimage | | |
|image |Content Connections from Previous Grades: 3.3, 7.2, 8.1 | |
| | |(Have students identify the shape or one ordered pair of a polygon |
|Symbols and Notation: | |rotated 90 degrees counterclockwise about the origin… or … a composite |
| | |transformation of a reflection and a translation or a reflection over y|
|T(x, y) → (x + 3, y – 2) | |= x and a dilation, etc) |
|“ T of x, y maps to x plus 3, y minus 2” | | |
|[pic] “x plus 3, y minus 2” | | |
|(x, y) → (2x, 2y) “x, y maps to 2x, 2y” | | |
|Core Standard: H.3G Geometry Score Reporting Category 2 |
|Transform and analyze figures. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.3G.3 Apply a scale factor to determine similar two- and |Items assessing this standard require students to use ratios and proportions |The Smiths want to build a new doghouse with a volume 4 times that of |
|three-dimensional figures, are similar. Compare and compute |to solve for areas and volumes of similar figures. Students may be given two|their old one. |
|their respective areas and volumes of similar figures. |surface areas and one volume for two similar figures and be asked to find the|What might the dimensions of the new doghouse be? |
| |volume of the second figure. |Shown is a picture of the original doghouse without the roof. |
|Assessable Academic Vocabulary: |Identify the scale factor between two similar figures. |[pic] |
| |Be able to understand formulas for surface area and volume. |7 ft. by 6.5 ft. by 6 ft. |
|scale factor |Write and solve algebraic equations. |2 ft. by 2.5 ft. by 6 ft. |
|similar figures |Square and cube numbers as well as find the square root and cube root of |3 ft. by 4 ft. by 5 ft. |
|area |numbers. |4 ft. by 5 ft. by 6 ft. |
|surface area |Answers may be exact or approximate. |ITS #100312 |
|volume | | |
| | |(Given the scale factor between two similar figures, find the ratio of |
| | |their areas.) |
|Symbols and Notation: | | |
| |Content Connections from Previous Grades: 3.3, 5.3, 7.2, 7.3 | |
|a : b “ratio of a to b” | |(Given the volumes of two similar figures and one of their surface |
| | |areas, find the surface area of the second figure.) |
|[pic] “ratio of a to b” | | |
|Core Standard: H.3G Geometry Score Reporting Category 2 |
|Transform and analyze figures. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.3G.4 Apply slope, distance, and midpoint formulas to solve |Items assessing this standard require students to solve word problems on a |(Given two endpoints, find the midpoint of a segment.) |
|problems in a coordinate plane. |coordinate grid, justify properties of two-dimensional shapes on a coordinate| |
| |grid, find midpoints or endpoints of segments among other problems related to| |
| |using slope, distance, and midpoint formulas for solving or justification | |
| |purposes. |(Given two points, find the distance between them…answer is not an |
|Assessable Academic Vocabulary: |Be able to understand formulas for slope, distance, and midpoint. |integer… or … show a quadrilateral is a parallelogram using slopes of |
| |Understand that distinct lines with equal slopes are parallel. Also |sides.) |
|slope |understand that slopes with a product of -1 are perpendicular. | |
|distance formula |Understand that Pythagorean Theorem or the distance formula can be used to | |
|midpoint formula |find the distance between two points on a coordinate plane. | |
| |Write and solve algebraic equations. |(Find the endpoint of a segment given one endpoint and a midpoint…. or…|
| |Answers may be exact or approximate. |give a word problem – a store is at one point and a library at another |
| | |– and ask what the distance is between the buildings.) |
|Symbols and Notation: | | |
| |Content Connections from Previous Grades: 8.1, 8.3 | |
|Core Standard: H.1S Data Analysis Score Reporting Category 3 |
|Analyze and interpret empirical data. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.1S.1 Given a context, determine appropriate survey methods,|Items assessing this standard include understanding survey methods. |(Given a context, decide population and/or questions for survey and how|
|analyze the strengths and limitations of a particular survey, |Specifically, given a context, students will: |to display its data and/or determine an appropriate survey method.) |
|observational study, experiment, or simulation, and the display|identify the population and questions needed to conduct an appropriate survey| |
|of its data. |determine appropriate survey methods (e.g., mail, phone, random phone calls, | |
| |etc) | |
|Assessable Academic Vocabulary: |analyze the strengths and limitations of a survey, observational study, | |
| |experiment or simulation. | |
|empirical data |analyze the strengths and limitations of the data display corresponding to a | |
|observational study |survey, observational study, experiment, or simulation. |(Given a context, decide how best to conduct an experiment or |
|experiment |Data displays include bar graphs, histograms, line graphs, stem-and-leaf |simulation to collect data and how to display its data.) |
|survey |plots, box-plots, and circle graphs. | |
|strength of a survey | | |
|limitation of a survey | | |
|data display |Content Connections from Previous Grades: | |
| |5.1.7, 8.2 |(Given a context, data, and display of data, analyze its strengths and |
| | |limitations. This may include asking why a display of data is |
| | |misleading in a particular context.) |
| | | |
| | | |
|Symbols and Notation: | | |
| | | |
| | | |
|Core Standard: H.1S Data Analysis Score Reporting Category 3 |
|Analyze and interpret empirical data. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.1S.2 Evaluate data-based reports by considering the source |Items assessing this standard evaluate data-based reports by considering one |(Given a data-based report and the source of the data, decide if the |
|of the data, the design of the study, and the way the data was |or more of the following: |data is biased.) |
|analyzed and displayed. |the source of the data | |
| |the design of the study | |
|Assessable Academic Vocabulary: |the way the data was analyzed | |
| |the way the data was displayed | |
|data display |Data displays include bar graphs, histograms, line graphs, stem-and-leaf | |
|evaluate |plots, box-plots, and circle graphs. |(Given two different displays of the same data, decide which tells the |
|source of data | |truth.) |
| | | |
| | | |
| | | |
| | | |
| | |(Given a box-and-whisker plot and some evaluative statement about the |
| |Content Connections from Previous Grades: |data, determine if the analysis is accurate.) |
| |5.1.7, 8.2 | |
|Symbols and Notation: | | |
| | | |
|Core Standard: H.1S Data Analysis Score Reporting Category 3 |
|Analyze and interpret empirical data. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.1S.3 Compare and draw conclusions about two or more data |Use graphical displays to compare and draw conclusions about two or more data|(Given a context and two sets of data, decide which graphical display |
|sets using graphical displays or central tendencies and range. |sets. |to use and why.) |
| |Graphical displays include bar graphs, histograms, line graphs, stem-and-leaf| |
|Assessable Academic Vocabulary: |plots, box-plots, and circle graphs. | |
| |Using mean, median, mode and/or range, compare and draw conclusions about two| |
|graphical display |or more data sets. | |
|measures of central tendencies | |(Given a context, two histograms (using two different intervals), what |
|mean | |conclusions can be drawn?) |
|median | | |
|mode | | |
|range | | |
|histogram |Content Connections from Previous Grades: |(Draw conclusions given two box-and-whisker plots.) |
|bar graph |5.1.7, 8.2 | |
|circle graph | | |
|line graph | | |
|box-plot | | |
|stem-and-leaf plot | | |
| | | |
| | | |
| | | |
|Symbols and Notation: | | |
|Core Standard: H.1S Data Analysis Score Reporting Category 3 |
|Analyze and interpret empirical data. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.1S.4 Use or construct a scatter plot for a given data set, |Items assessing this standard include the following: |(Identify a scatter plot that matches the data in a given data set.) |
|determine whether there is a(n) linear, quadratic, exponential,|Construct a scatter plot given a data set. | |
|or no trend. If linear, determine if there is a positive or |Use a scatter plot to determine whether the data shows a(n) linear, | |
|negative correlation among the data; and, if appropriate, |quadratic, exponential or no trend. | |
|sketch a line of best fit, and use it to make predictions. |Use a scatter plot with a linear trend to determine if there is a positive or| |
| |negative correlation. Students may have to identify the line of best fit |(Identify whether data in a scatter plot is linear, quadratic, |
|Assessable Academic Vocabulary: |graphically or algebraically (write an equation). Students may have to use |exponential, or has no trend. (or identify a line of best fit on a |
| |their line to make predictions related to the data. |scatter plot – graphically) If linear, determine if there is a |
|scatter plot |Scatter plots may be drawn in any quadrant on a coordinate plane. |positive or negative correlation among the data.) |
|linear trend |Students may need to consider whether their data shows a strong or weak | |
|quadratic trend |correlation. | |
|exponential trend | | |
|no trend | |(Find an equation for a line of best fit or make predictions about a |
|positive correlation |Content Connections from Previous Grades: 5.1.7, 8.2 |data set using a line of best fit.) |
|negative correlation | | |
|line of best fit | | |
| | | |
| | | |
| | | |
|Symbols and Notation: | | |
|Core Standard: H.2S Probability Score Reporting Category 3 |
|Apply basic principles of probability. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.2S.1 Identify, analyze, and use experimental and |Items assessing this standard include using experimental probability and |(Given the experimental probability of Cal hitting a home run when he |
|theoretical probability to estimate and calculate the |theoretical probability to estimate the probability of simple events. |is at bat is 0.2, estimate the number of times Cal hits a home run if |
|probability of simple events. |Compute simple theoretical probability. |he is at bat 4 times in a game.) |
| |Compute simple experimental probability given necessary values. | |
|Assessable Academic Vocabulary: |Analyze estimates and calculations of theoretical and experimental | |
| |probabilities. A student may need to find an experimental probability and use| |
|probability |proportions or calculate the number of times something is likely to occur |(Karen spins a spinner that has four colors – red, blue, yellow, green |
|experimental probability |given a probability. |– each ¼ of the spinner. If she spins the spinner 20 times, estimate |
|theoretical probability |Determine the size of the sample space needed to calculate probabilities. |the number of times she will spin blue.) |
|simple event |Answers may be exact or approximated. | |
| | | |
| | | |
|Symbols and Notation: | | |
| |Content Connections from Previous Grades: 6.2, 7.2, 8.2 | |
|P(Event) “Probability of Event” | |(Tina counts 36 deer and tags them in a forest. She goes back 6 months|
| | |later and sees 70 deer, 10 of which are tagged. Estimate the number of|
| | |deer in the forest.) |
|Core Standard: H.2S Probability Score Reporting Category 3 |
|Apply basic principles of probability. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.2S.2 Determine the sample space of a probability |Items assessing this standard include listing a sample space as well as |(A student flips a coin two times. What is the sample space?) |
|experiment. |determining the number of items in the sample space. | |
| |Solutions may require the multiplication counting principle. | |
| |Make lists to determine the sample space. | |
| |Understand the relationship between the sample space and the probability of | |
|Assessable Academic Vocabulary: |an event. | |
| | |(A student spins a spinner with three colors – red, blue, and green – |
|sample space | |two times. What is the sample space?) |
| | | |
| | | |
| |Content Connections from Previous Grades: 6.2, 7.2 | |
|Symbols and Notation: | | |
| | |(A bag has four marbles. Two are blue, one green, one red. What is |
|{ heads, tails } “the set heads, tails” | |the sample space for choosing two marbles with replacement?) |
|Core Standard: H.2S Probability Score Reporting Category 3 |
|Apply basic principles of probability. |
|It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. |
|Content Standard: |Boundaries of Assessable Content: |Sample Items: |
|H.2S.3 Compute and interpret probabilities for independent, |Items assessing this standard will include various methods to compute |If the spinner shown is spun, what is the probability it would NOT land|
|dependent, complementary, and compound events using various |probabilities. Students may use the various methods learned to solve the |on 3? |
|methods (e.g., diagrams, tables, area models, and counting |problem or may be asked to interpret a method and explain why it shows the |[pic] |
|techniques). |requested probability. |A. B. C. D. |
| |A student may have to find the probability of an event or be given the | |
|Assessable Academic Vocabulary: |probability of an event and use that to determine the probability of the | |
| |complementary event. |ITS #100598 |
|independent events |Counting techniques include multiplication counting rule and may include some|Tim rolls two six-sided dice. |
|dependent events |combinations and permutations. |Given the sum is odd, what is the probability the sum is 5? |
|complement | |A. B. C. D. |
|complementary events | |ITS #100590 |
|compound events | | |
|area models | | |
|counting techniques |Content Connections from Previous Grades: 6.2, 7.2 |A pair of six-sided dice is rolled, and the result is a sum less than |
|with replacement | |10. |
|without replacement | |What is the probability both dice have the same numerical value? |
| | |A. B. C. D. |
|Symbols and Notation: | | |
| | |ITS #100601 |
|P(Event) “Probability of Event” | | |
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